Average Error: 0.1 → 0.1
Time: 18.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r514607 = e;
        double r514608 = v;
        double r514609 = sin(r514608);
        double r514610 = r514607 * r514609;
        double r514611 = 1.0;
        double r514612 = cos(r514608);
        double r514613 = r514607 * r514612;
        double r514614 = r514611 + r514613;
        double r514615 = r514610 / r514614;
        return r514615;
}
↓
double f(double e, double v) {
        double r514616 = e;
        double r514617 = v;
        double r514618 = sin(r514617);
        double r514619 = cos(r514617);
        double r514620 = 1.0;
        double r514621 = fma(r514619, r514616, r514620);
        double r514622 = r514618 / r514621;
        double r514623 = r514616 * r514622;
        return r514623;
}

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot e}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.1

    \[\leadsto \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \color{blue}{\left(1 \cdot e\right)}\]

  5. Applied associate-*r*0.1

    \[\leadsto \color{blue}{\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot 1\right) \cdot e}\]

  6. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\]

  7. Final simplification0.1

    \[\leadsto e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))